Hecke modules based on involutions in extended Weyl groups

Abstract

Let X X be the group of weights of a maximal torus of a simply connected semisimple group over C \mathbf {C} and let W W be the Weyl group. The semidirect product W ( ( Q ⊗ X ) / X ) W((\mathbf {Q}\otimes X)/X) is called an extended Weyl group. There is a natural C ( v ) \mathbf {C}(v) -algebra H \mathbf {H} called the extended Hecke algebra with basis indexed by the extended Weyl group which contains the usual Hecke algebra as a subalgebra. We construct an H \mathbf {H} -module with basis indexed by the involutions in the extended Weyl group. This generalizes a construction of the author and Vogan.